An enhanced formulation of error bound in subspace iteration method - Chen, Pu; Gong, Yucai; Yongqiang Chen; Kulasegaram, Sivakumar (International Journal for Numerical Methods in Biomedical EngineeringInternational Journal for Numerical Methods in Biomedical Engineering, 2011)
In subspace iteration method (SIM), the relative difference of approximated eigenvalues between two consecutive iterations is usually employed as the convergence criterion. However, though it controls the convergence of eigenvalues well, it cannot guarantee the convergence of eigenvectors in all cases. In the case when there is no shifting, the best choice for the convergence criterion of eigenvalues may generally be the computable error bound proposed by Matthies, which is based on an estimation of Rayleigh quotient of approximated eigenvalues expressed in the subspace. Matthies' form guarantees the convergence of both eigenvalues and eigenvectors and can be computed with almost negligible operations. However, it is not as popular as expected in implementations, partly because it does not consider the popular shifting acceleration technique of subspace iterations. In this paper, we extend Matthies' form to the case of nonzero shifting and prove that this extended error bound form with nonzero shifting can be used generally as a convergence criterion for eigenpairs. Besides, this paper details the derivation to illustrate that the extended error bound can also be applied to the case of positive semi-definite mass matrix by only slightly modifying the subspace iteration procedure. Numerical tests are presented to illustrate the motivation and to demonstrate the better performance of the modified computable error bound. The studies in this paper indicate that the modified Matthies' form of error bound can be effectively used as a preferred convergence criterion in the SIM. Copyright (C) 2009 John Wiley & Sons, Ltd.
Keywords: convergence criteria; error bounds; finite element; subspace iteration method